arXiv subjects

Dominic Wynter

Publications and source records attributed to Dominic Wynter.

4 recordsLinked to original sources

arXiv2025

From the quantum Boltzmann operator to the quantum Landau operator

In this manuscript we derive the quantum Landau operator as the weak-coupling limit of the quantum Boltzmann operator (also known as the Uehling-Uhlenbeck operator). We consider both Fermi-Dirac and Bose-Einstein statistics. Our approach is inspired by the work by Benedetto and Pulvirenti, where the classical Landau operator was derived from the quantum Boltzmann operator. To capture the ternary term in the quantum Landau operator, we introduce a new two-parameter scaling that preserves the quantum effects in the limit. Furthermore, we provide an explicit rate of convergence that depends on the regularity of the interaction potential.

math.AP↗

arXiv2024

Shock profiles for the non-cutoff Boltzmann equation with hard potentials

The Boltzmann equation models gas dynamics in the low density or high Mach number regime, using a statistical description of molecular interactions. Planar shock wave solutions have been constructed for the Boltzmann equation for hard potentials with angular cutoff, and more recently for the Landau equation of plasma dynamics. In this work, we construct shock profile solutions for the Boltzmann equation where the molecular interactions are long-range, and we show these solutions to be smooth and well approximated by compressible Navier Stokes shock profiles. Our proof procedes by standard energy estimates and a quantitative Chapman-Enskog approximation.

math.AP↗

arXiv2023

Global Well-Posedness and Large Data Estimates for the 1D Boltzmann Equation

We prove quantitative growth estimates for large data solutions to the 1D Boltzmann equation, for a collision kernel with angular cutoff and relative velocity cutoff. We present proofs for the global well-posedness results presented in the note of Biryuk, Craig, and Panferov, in which global solutions for this equation are shown to exist for large data, with density bounded for all time. We show that these solutions propagate moments in $v$, and derivatives in $x$ and $v$. Our main contribution is to develop new, sharp integral inequality estimates, which allow us to prove exponential growth bounds in $L^\infty_x L^1_v$ for large data, and to prove dissipation in $ L^\infty_x L^1_v$ for finite energy data on the line.

math.AP↗

arXiv2021

Quantitative Propagation of Chaos for the Mixed-Sign Viscous Vortex Model on the Torus

We derive a quantiative propagation of chaos result for a mixed-sign point vortex system on $\mathbb{T}^2$ with independent Brownian noise, at an optimal rate. We introduce a pairing between vortices of opposite sign, and using the vorticity formulation of 2D Navier-Stokes, we define an associated tensorized vorticity equation on $\mathbb{T}^2\times\mathbb{T}^2$ with the same well-posedness theory as the original equation. Solutions of the new PDE can be projected onto solutions of Navier-Stokes, and the tensorized equation allows us to exploit existing propagation of chaos theory for identical particles.

math-ph↗